Categories

# variational principles in classical mechanics solutions

The following examples of variable mass systems illustrate subtle complications that occur handling such problems using algebraic mechanics. This book provides a systematic, highly practical introduction to the use of energy principles, traditional variational methods, and the finite element method for the solution of engineering problems involving bars, beams, torsion, plane elasticity, trusses, and plates. The first scenario is the "folded chain" system which assumes that one end of the chain is held fixed, while the adjacent free end is released at the same altitude as the top of the fixed arm, and this free end is allowed to fall in the constant gravitational field $$g$$. Mathematical evaluation of the variational principles 7. It is especially useful for solving motion in rotating systems in science and engineering. This result is very different from that obtained using the erroneous assumption that the right arm falls with the free-fall acceleration $$g$$, which implies a maximum tension $$T_{0}=$$ $$2Mg$$. Important applications of Hamiltonian mechanics are to quantum mechanics and statistical mechanics, where quantum analogs of $$q_{i}$$ and $$p_{i},$$ can be used to relate to the fundamental variables of Hamiltonian mechanics. The "falling chain", scenario assumes that one end of the chain is hanging down through a hole in a frictionless, smooth, rigid, horizontal table, with the stationary partition of the chain lying on the frictionless table surrounding the hole. C.G. This book introduces variational principles and their application to classical mechanics. Using an idealized one-dimensional assumption, the Lagrangian $$\mathcal{L}$$ is given by $\mathcal{L}(y,\dot{y})=\frac{M}{4L}(L-y)\dot{y}^{2}+Mg\frac{1}{4L} (L^{2}+2Ly-y^{2})$ where the bracket in the second term is the height of the center of mass of the folded chain with respect to the fixed upper end of the chain. A light (massless) spring of spring constant k is attached between the two particle. In both examples, the chain, with mass $$M$$ and length $$L,$$ is partitioned into a stationary segment, plus a moving segment, where the mass per unit length of the chain is $$\mu =\frac{M}{L}$$. The Variational Principles of Mechanics Addeddate 2016-10-20 08:43:09 Identifier Two dramatically different philosophical approaches to classical mechanics were proposed during the 17th – 18th centuries. Alternatively, use of Lagrange multipliers allows determination of the constraint forces resulting in $$n+m$$ second order equations and unknowns. energy-principles-and-variational-methods-in-applied-mechanics 1/3 Downloaded from calendar.pridesource.com on November 12, 2020 by guest Read Online Energy Principles And Variational Methods In Applied Mechanics Right here, we have countless ebook energy principles and variational methods in applied mechanics and collections to check out. Two Routhians are used frequently for solving the equations of motion of rotating systems. Just as in quantum mechanics, variational principles can be used directly to solve a dynamics problem, without employing the equations of motion. Clebsch variational principles in ﬁeld theories and singular solutions of covariant EPDiﬀ equations Francois Gay-Balmaz1 Abstract This paper introduces and studies a ﬁeld theoretic analogue of the Clebsch variational principle of classical mechanics. As shown in the discussion of the Generalized Energy Theorem, (chapters $$8.8$$ and $$8.9$$), when all the active forces are included in the Lagrangian and the Hamiltonian, then the total mechanical energy $$E$$ is given by $$E=H.$$ Moreover, both the Lagrangian and the Hamiltonian are time independent, since $\frac{dE}{dt}=\frac{dH}{dt}=-\frac{\partial \mathcal{L}}{\partial t}=0$ Therefore the "folded chain" Hamiltonian equals the total energy, which is a constant of motion. By contrast, for the "falling system", the chain links are transferred from the stationary upper section to the moving lower segment of the chain. The variational approach to mechanics 2. The fixed end is attached to a fixed support while the free end of the chain is dropped at time $$t=0$$ with the free end at the same height and adjacent to the fixed end. This is not possible using the Lagrangian approach since, even though the $$m$$ coordinates $$q_{i}$$ can be factored out, the velocities $$\dot{q}_{i}$$ still must be included, thus the $$n$$ conjugate variables must be included. The goal of this book is to introduce the reader to the intellectual beauty, and philosophical implications, of the fact that nature obeys variational principles that underlie the Lagrangian and Hamiltonian analytical formulations of classical mechanics. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. For a system with $$n$$ generalized coordinates, plus $$m$$ constraint forces that are not required to be known, then the Lagrangian approach, using a minimal set of generalized coordinates, reduces to only $$s=n-m$$ second-order differential equations and unknowns compared to the Newtonian approach where there are $$n+m$$ unknowns. First and foremost, the entire theory of Variational principles in fluid dynamics may be divided into two categories. Variational Principles in Classical Mechanics by Douglas Cline is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0), except where other-wise noted. Assuming that the variables between $$1\leq i\leq s$$ are non-cyclic, while the $$m$$ variables between $$s+1\leq i\leq n$$ are ignorable cyclic coordinates, then the two Routhians are: \begin{aligned} R_{cyclic}(q_{1},\dots ,q_{n};\dot{q}_{1},\dots ,\dot{q}_{s};p_{s+1},\dots .,p_{n};t) &=&\sum_{cyclic}^{m}p_{i}\dot{q}_{i}-L=H-\sum_{noncyclic}^{s}p_{i}\dot{q}_{i} \label{8.65} \\ R_{noncyclic}(q_{1},\dots ,q_{n};p_{1},\dots ,p_{s};\dot{q}_{s+1},\dots .,\dot{q} _{n};t) &=&\sum_{noncyclic}^{s}p_{i}\dot{q}_{i}-L=H-\sum_{cyclic}^{m}p_{i} \dot{q}_{i} \label{8.68}\end{aligned}. Thus it is natural to compare the relative advantages of these two algebraic formalisms in order to decide which should be used for a specific problem. Since this moving section is falling downwards, and the stationary section is stationary, then the transferred momentum is in a downward direction corresponding to an increased effective downward force. The Routhian $$R_{noncyclic}$$ is a Hamiltonian for the non-cyclic variables between $$1\leq i\leq s$$, and is a negative Lagrangian for the $$m$$ cyclic variables between $$s+1\leq i\leq n$$. However, such systems still can be conservative if the Lagrangian or Hamiltonian include all the active degrees of freedom for the combined donor-receptor system. Thus the $$m$$ conjugate variables $$\left( q_{i},p_{i}\right)$$ can be factored out of the Hamiltonian, which reduces the number of conjugate variables required to $$n-m$$. These partitions are strongly coupled at their intersection which propagates downward with time for the "folded chain" and propagates upward, relative to the lower end of the falling chain, for the "falling chain". In this chapter we will look at a very powerful general approach to ﬁnding governing equations for a broad class of systems: variational principles. Kotkin's "Collection of Problems in Classical Mechanics": Last but not least, filling in the "with a lot of exercises" hole, Serbo & Kotkin's book is simply the key to score 101 out of 100 in any Mechanics exam. The maximum tension was $$\simeq$$ $$25Mg,$$ which is consistent with that predicted using Equation \ref{8.77} after taking into account the finite size and mass of individual links in the chain. Missed the LibreFest? The falling-mass system is conservative assuming that both the donor plus the receptor body systems are included. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The authors then launch into an analysis of their most significant topics: the relation between variational principles and wave mechanics, and the principles of Feynman and Schwinger in quantum mechanics. This book introduces variational principles and their application to classical mechanics. Newton developed his vectorial formulation that uses time-dependent differential equations of motion to relate vector observables like force and rate of change of momentum. Hamiltonian dynamics also has a means of determining the unknown variables for which the solution assumes a soluble form. Let $$y$$ be the distance the falling free end is below the fixed end. In elastostatics, particularly for problems involving random composites, the variational principle of Hashin and Shtrikman [3-5] has displayed a clear advantage over the classical energy principles, to which (2.9), (2.11) and (2.12) are analogous. Agenda 1 Variational Principle in Statics 2 Variational Principle in Statics under Constraints 3 Variational Principle in Dynamics 4 Variational Principle in Dynamics under Constraints Shinichi Hirai (Dept. There is hardly a branch of the mathematical sciences in which abstract rigorous speculation and experimental … That is, these partitions share time-dependent fractions of the total chain mass. The Lagrangian potential function is limited to conservative forces, Lagrange multipliers can be used to handle holonomic forces of constraint, while generalized forces can be used to handle non-conservative and non-holonomic forces. Consider two particles of masses m 1, and m 2. This principle yields an alternative Euler, Lagrange, Hamilton, and Jacobi, developed powerful alternative variational formulations based on the … This is in contrast to that for the folded chain system where the acceleration exceeds $$g$$. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. For the "falling- chain" let $$y$$ be the falling distance of the lower end of the chain measured with respect to the table top. Newtonian mechanics was used to solve the rocket problem in chapter $$3.12$$. This ability is impractical or impossible using Newtonian mechanics. Hamilton derived the canonical equations of motion from his fundamental variational principle and made them the basis for a far-reaching theory of dynamics. [ "article:topic", "authorname:dcline", "license:ccbyncsa", "showtoc:no" ]. These partitions are coupled at the moving intersection between the chain partitions. The Classical Variational Principles of Mechanics J. T. Oden 1.1 INTRODUCTION The last twenty years have been marked by some of the most significant advances in variational mechanics of this century. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Since the cyclic variables are constants of the Hamiltonian, their solution is trivial, and the number of variables included in the Lagrangian is reduced from $$n$$ to $$s=n-m$$. Thus energy conservation can be used to give that $E=\frac{1}{2}\mu y(\dot{y}^{2}-gy)=E_{0}$ Lagrange’s equation of motion gives $\dot{p}_{y}=m_{y}\ddot{y}+\dot{m}_{y}\dot{y}=m_{y}g+\frac{1}{2}\mu \dot{y} ^{2}=Mg-T_{0}$. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The equation of motion for rocket motion is easily derived using either Lagrangian or Hamiltonian mechanics by relating the rocket thrust to the generalized force $$Q_{j}^{EXC}.$$, The motion of a flexible, frictionless, heavy chain that is falling in a gravitational field, often can be split into two coupled variable-mass partitions that have different chain-link velocities. Lagrangian and the Hamiltonian dynamics are two powerful and related variational algebraic formulations of mechanics that are based on Hamilton’s action principle. Since the cyclic variables are constants of motion, the Routhian $$R_{noncyclic}$$ also is a constant of motion but it does not equal the total energy since the coordinate transformation is time dependent. PHYS 316: Advanced Classical Mechanics.